Convexification of electrical impedance tomography with restricted Dirichlet-to-Neumann map data

Abstract

We propose a new numerical method to reconstruct the isotropic electrical conductivity from measured restricted Dirichlet-to-Neumann map data in electrical impedance tomography (EIT). ‘Restricted Dirichlet-to-Neumann (DtN) map data’ means that the Dirichlet and Neumann boundary data for EIT are generated by a point source running either along an interval of a straight line or along a curve located outside of the domain of interest. We ‘convexify’ the problem via constructing a globally strictly convex Tikhonov-like functional using a Carleman weight function. In particular, two new Carleman estimates are established. Global convergence to the correct solution of the gradient projection method for this functional is proven. Numerical examples demonstrate a good performance of this numerical procedure.